Numerical Integration Method Research Articles

Automation of the technological process of forming details with straight rifts

The subject matter of the article is technological line for the production of thick sheet parts with rifts, made of structural and special steels. The goal of the work is analysis and creation of methods of managing basic technological operations at all stages of the technological process of the automated production of such parts. The following tasks were solved in the article: formation of models of the main stages of forming parts with rectilinear rifts and control of dynamic properties of selected systems of their implementation. The following methods used are – methods of numerical integration and numerical methods of solving differential equations; analytical solution of differential equations based on the Dahlembert principle or by using the Lagrange equation; frequency analysis of transfer functions of individual links, presented in the form of fractional-rational functions. The following results were obtained – constructed in the first approximation the composition and sequence of the main operations of the generalized technological process of the automated production of thick-walled sheet parts with rifts; according to the results of the analysis of the process of elastic-plastic change of the shape of the region with many connections, the input and output parameters necessary for building a control model of the specified process were determined; it is determined that for its implementation it is necessary to ensure free deformation of at least 0.25 % of the rift surface area outside of contact with the forming tool, and the necessary technological methods for this are proposed; proposed variants of the technical implementation of the selected technological methods; the results of modeling schemes and modes of operation of the necessary technical equipment are presented. Conclusions: The application of the proposed structure, composition and sequence of main operations of the generalized technological process of the automated production of thick-walled sheet parts with cracks, based on its implementation by the selected methods on the proposed equipment, will allow to ensure the effective management of the quality of the production of the specified parts at each stage of the technological process.

ESBD: Exponential Strain-based Dynamics using XPBD algorithm

Strain-based Dynamics (SBD) (Müller et al., 2014) is a well-known method to simulate the deformable models by using Green’s strain tensor within the Position-based Dynamics (PBD) algorithm. Similar to the other position-based constraints, SBD performs stable and robust. However, when we transform the base algorithm from PBD to Extended Position-based Dynamics (XPBD), SBD suffers from the cumbersome parameter adjustments and uncontrollable numerical dissipation, which generates highly stiff results. To overcome these limitations, we propose Exponential Strain-based Dynamics (ESBD) which reformulates the Green’s strain tensor with an exponential treatment. Our proposed method is less sensitive to the increased numerical dissipation, so it provides vivid simulation of deformable objects with fewer parameter tunings. In order to present the advantages, we compare our method with SBD by using the XPBD method and other well-known techniques. We observe that ESBD improves the hyperelastic behavior of the Green’s strain tensor as a projected constraint without altering the underlying XPBD algorithm. Therefore, it is possible to adapt ESBD to the other existing numerical integration and iteration methods. Our proposed method is stable, straightforward to follow and produces high quality dynamic simulations.

Study of the Effect of Some Muzzle Device Types on the Firing Force and Firing Impulse

In this article, a mathematical model is used to determine the force and impulse of the weapon’s firing. The force and impulse of the weapon’s firing are determined during the whole firing process, which consists of two stages: the stage projectile moving inside the barrel and the stage at the final action stage of combustion gas. The input parameters of the UK-59 machine gun in the Czech Republic were selected to verify the mathematical model. The results of the article are the law of combustion gas pressure in the barrel, the law of combustion gas pressure in the gas chamber, and the law of force and impulse of the shot corresponding to different types of muzzle devices. This result is consistent with the experimental results presented in the document, and the error is less than 3%. Besides, the article also analyzes the influence of some muzzle device types on the firing force and firing impulse. The method of numerical integration has been applied to solve the problem using Matlab software. The research results of the article are a reliable theoretical basis for determining the firing forces, serving the problem of designing the mount as well as the firing stability of the weapon when firing. This is the basis for the optimal selection and design of the muzzle device used in the weapon. Received: 14 July 2023 | Revised: 1 September 2023 | Accepted: 20 September 2023 Conflicts of Interest The authors declare that they have no conflicts of interest to this work. Data Availability Statement Data available on request from the corresponding author upon reasonable request. Author Contribution Statement Bien Vo Van: Conceptualization, Methodology, Software, Validation, Formal analysis, Investigation, Data curation, Writing - original draft, Writing - review & editing, Visualization, Supervision, Project administration; Phon Nguyen Duy: Software, Formal analysis, Resources, Writing - review & editing; Phu Nguyen Minh: Methodology, Software, Validation, Formal analysis, Investigation, Writing - original draft, Writing - review & editing.

Kernel method for estimating overlapping coefficient using numerical integration methods

In this paper, we proposed three nonparametric kernel estimators for the overlapping Weitzman measure Δ. Due to the difficulty of finding a general expression for Δ when the nonparametric kernel method is adopted, we suggest using the numerical integration method as the first stage of our estimation process. In particular, three numerical integration rules are considered, which are known as, trapezoidal and Simpson rules. The statistical properties of the resulting estimators are studied and investigated by using the simulation technique and their performances are also compared with some existing nonparametric kernel estimators developed by Eidous and Al-Talafhah (2020). The numerical results demonstrated the superiority and usefulness of the proposed technique over that suggested by Eidous and Al-Talafhah in estimating the overlapping measure Δ.

Spectral method of electrical circuits accelerated simulation with thyristors

Purpose. The development of transient processes calculation method in electric circuits with thyristors based on the use of functions approximation by orthogonal polynomials.
 Methodology. Functions approximation by orthogonal polynomials, numerical methods of differential equations integration, matrix methods, programming, theory of electric circuits.
 Obtained results. The method of solution function polynomial approximation of integro-differential equations of state, which describes the transient processes of an electric circuit with thyristors, is used in this paper. The used method showed the advantages over other known methods in increasing the accuracy and reducing the simulation time of transient electrical processes by more than 6 times.
 Findings. The solution is approximated by a series of Chebyshev polynomials. The integro-differential equations of state are transformed into linear algebraic equations for special depiction of the solution functions. The depiction of functions of true currents in the equivalent circuit is interpreted as direct currents. Such a schematic model creates visibility for a researcher performing simulation of transient electrical processes.
 Practical value. The proposed methods discover the possibility of using the apparatus of direct current electric circuits’ theory for transient processes in complex schemes modeling with thyristors.

Charge instability of JMaRT geometries

We perform a detailed study of linear perturbations of the JMaRT family of non-BPS smooth horizonless solutions of type IIB supergravity beyond the near-decoupling limit. In addition to the unstable quasi normal modes (QNMs) responsible for the ergo-region instability, already studied in the literature, we find a new class of ‘charged’ unstable modes with positive imaginary part, that can be interpreted in terms of the emission of charged (scalar) quanta with non zero KK momentum. We use both matched asymptotic expansions and numerical integration methods. Moreover, we exploit the recently discovered correspondence between JMaRT perturbation theory, governed by a Reduced Confluent Heun Equation, and the quantum Seiberg-Witten (SW) curve of N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 2 SYM theory with gauge group SU(2) and Nf = (0, 2) flavours.

A new symplectic integrator for stochastic Hamiltonian systems on manifolds

This paper presents a significant development in the field of geometric numerical stochastic integration schemes. Specifically, the Geometric Symplectic Itô-Taylor 1.0 strong numerical integration scheme, tailored for Hamiltonian systems evolving on 2-sphere or S2 manifold is proposed. The core of this proposed algorithm centers around a new weak symplectic condition. This condition ensures the numerical stability for long duration time integration that can be readily achieved through numerical steps rather than relying solely on analytical satisfaction of the condition. This proposed advancement over the state-of-the-art caters to improved accuracy and stability of numerical simulations in manifold-based dynamical systems for long duration simulations. Through extensive analysis and numerical experiments, the effectiveness and reliability of the proposed scheme and symplectic criterion are validated. The findings of this study promises to offer valuable insights for researchers and practitioners working in the field of symplectic numerical integration methods.

Vibration characteristics of linear and nonlinear dissipative elastic metamaterials rotor with geometrical nonlinearity

Theoretical researches on the vibration characteristics of a dissipative elastic metamaterials rotor with geometrical nonlinearity are presented in this work. Both linear and nonlinear damping cases are implemented in parallel to the spring of the unit. The dynamic model of the dissipative metamaterials rotor is built and discretized by the assumed mode method. Then the solutions of the nonlinear equations are obtained using the harmonic balance method, and the numerical integration method is applied to validate the solutions obtained by the analytical method. The comparisons of the dynamic behaviors for the rotor with different types of geometrical nonlinearities and dissipations are presented, and the effect of the distribution position of the absorber unit on the dynamic behaviors is investigated. The results show that the linear damping has a better capacity to reduce the amplitude compared to the nonlinear damping due to the energy dissipation near the resonance region. On the other hand, compared to the linear dissipation, the nonlinear dissipation has a better performance around the bandgap that forbids the propagation of the vibration. Moreover, these properties can be further improved by adjusting the position of the unit on the rotor. The results presented in this paper provide a theoretical basis for enhancing the properties of metamaterials rotors.

Implicit–explicit multirate infinitesimal stage-restart methods

Implicit–Explicit (IMEX) methods are flexible numerical time integration methods which solve an initial-value problem (IVP) that is split into stiff and nonstiff processes with the goal of lower computational costs than a purely implicit or explicit approach. A complementary form of flexible IVP solvers are multirate infinitesimal methods for problems split into fast- and slow-changing dynamics, that solve a multirate IVP by evolving a sequence of “fast” IVPs using any suitably accurate algorithm. This article introduces a new class of high-order implicit–explicit multirate methods that are designed for multirate IVPs in which the slow-changing dynamics are further split in an IMEX fashion. This new class, which we call implicit–explicit multirate infinitesimal stage-restart (IMEX-MRI-SR), both improves upon the previous implicit–explicit multirate infinitesimal generalized-structure additive Runge Kutta (IMEX-MRI-GARK) methods by allowing for far easier creation of new embedded methods, and extends multirate exponential Runge Kutta (MERK) methods by allowing the fast-changing dynamics to be nonlinear and the methods to be implicit. We leverage GARK theory to derive conditions for orders of accuracy up to four, and we provide second- and third-order accurate example methods, which are the first known embedded MRI methods with IMEX structure. We then perform numerical simulations demonstrating convergence rates and computational performance in both fixed-step and adaptive-step settings.

A Novel All-Digital Resolver-to-Digital Conversion System Based on Numerical Synchronous Integration

Resolvers are universally employed as absolute angle sensors in motor control systems. Detecting the synchronous envelopes is an essential step during the Resolver-to-Digital Conversion (RDC) process when calculating the angle. However, conventional filter-based envelope detection could introduce enormous angle delay and peak capturing-based detection could lead to apparent quantization angle error. A novel all-digital RDC system is proposed in this paper with the synchronous envelopes detected based on numerical synchronous integration method. The sine and cosine signals are oversampled and then multiplied by the excitation signal before integration to derive the envelopes. The angle delay and quantization errors could thus be largely reduced compared with RDC systems based on conventional envelop detections. In the experiment, angle error is limited to less than 30 arc min at the angular speed of 1000 rpm.

Dynamic response analysis of a typical time-varying mass system: A moving-interface pipe model, the WKB-recursive solution and experimental validation

This research aims to seek a suitable way for analyzing the time-varying characteristics of some engineering phenomena such as the fuel consumption of flying rockets, the movement process of control rods used to regulate the reaction rate in nuclear reactors, etc. To address this problem, a simulated procedure containing modeling and response solving is proposed. A common variable cross-section pipeline with time-varying mass is designed as a practical time-varying system. A simplified segmented beam model with a moving interface is proposed to simulate the pipe's time-varying behavior. Then the Wentzel-Kramers-Brillouin (WKB)-recursive method is proposed to solve this time-varying problem. A two-segmented beam example is used to verify its computability. The computational efficiency is greatly improved in comparison with the conventional numerical integration methods. To validate this proposed procedure, numerical simulation is carried out and an experiment is specially designed and implemented. In the experiment, many cases of different rates of mass variation and excitation forces are carried out. Overall, the numerical dynamic responses match well with the experimental ones, which indicates that the proposed procedure is suitable for analyzing the system's time-varying characteristics.

ВДОСКОНАЛЕНА МАТЕМАТИЧНА МОДЕЛЬ ТРИОБМОТКОВОЇ АСИНХРОННОЇ МАШИНИ

In phase coordinates, a mathematical model of a three-winding asynchronous machine with two windings on the stator as a structural element of electric drives and asynchronous generators with inverter and inverter-capacitor excitation for autonomous electric power plants, oriented to explicit methods of numerical integration of the system of differential equations, has been developed. The model takes into account the displacement angle between two stator windings in the range of 360º and other important factors affecting the course of electromagnetic processes occurring in the machine. On the basis of the mathematical model, a program code was developed as a means of computer verification of machine operation modes. The results of computer study performed with the help of the code confirm the suitability of the mathematical model of a three-winding asynchronous machine to take into account the influence of the displacement angle between the stator windings and their simultaneous and single power supply during the operation of the machine in motor mode. References 11, figures 10.

Enhancing neurodynamic approach with physics-informed neural networks for solving non-smooth convex optimization problems

This paper proposes a deep learning approach for solving non-smooth convex optimization problems (NCOPs), which have broad applications in computer science, engineering, and physics. Our approach combines neurodynamic optimization with physics-informed neural networks (PINNs) to provide an efficient and accurate solution. We first use neurodynamic optimization to formulate an initial value problem (IVP) that involves a system of ordinary differential equations for the NCOP. We then introduce a modified PINN as an approximate state solution to the IVP. Finally, we develop a dedicated algorithm to train the model to solve the IVP and minimize the NCOP objective simultaneously. Unlike existing numerical integration methods, a key advantage of our approach is that it does not require the computation of a series of intermediate states to produce a prediction of the NCOP. Our experimental results show that this computational feature results in fewer iterations being required to produce more accurate prediction solutions. Furthermore, our approach is effective in finding feasible solutions that satisfy the NCOP constraint.

Performance Improving of Wind Power Generation Systems Through Parameter Optimization and Dynamic Analysis of the Speed-Regulating Differential Transmission

Abstract Hybrid drive wind power generation systems (WPGSs) equipped with speed-regulating differential mechanisms (SRDMs) have emerged as a promising solution for integrating large-scale wind energy into the power grid without the need for partially or fully rated converters. This article presents a comprehensive study on the dynamic analysis and parameter optimization of the SRDM-based transmission, with the aim of providing a sound foundation for the design and performance improvement of hybrid drive WPGSs. This study first formulates the kinematics, power flow, and mechanical efficiency of the SRDM and then proposes an effective parameter configuration model for optimizing the speed ratios of the key link units. The objective function is set as the minimum peak power required for speed regulation by the SRDM. Furthermore, to deal with the unique mechanical features such as dual power inputs, continuously variable transmission, and time-varying steering mechanism, an appropriate nonlinear dynamic modeling method of the SRDM transmission is developed. The torsion–translation vibration equations are derived and solved using the Runge–Kutta numerical integral method, considering randomly changing wind speed inputs and time-varying internal/external excitations. Results reveal that the sun gear experiences severe vibrations with the maximal and average vibration displacements of 0.563 mm and 0.112 mm, respectively, in the circumferential direction, while the planet gear exhibits complex frequency responses. Finally, specialized case studies are demonstrated to verify the proposed approaches, showing the satisfactory on-grid operating performance of the proposed SRDM-based WPGSs.

Comparison of numerical-integration-based methods for blood flow estimation in diffuse correlation spectroscopy

Background and ObjectiveDiffuse correlation spectroscopy (DCS) is an optical blood flow monitoring technology that has been utilized in various biomedical applications. In signal processing of DCS, nonlinear fitting of the experimental data and the theoretical model can be a hindrance in real-time blood flow monitoring. As one of the approaches to resolve the issue, INISg1, the inverse of numerical integration of squared g1 (a normalized electric field autocorrelation function), that could surpass the state-of-the-art technique at the time in terms of signal processing speed, has been introduced. While it is possible to implement INISg1 using various numerical integration methods, no relevant studies have been performed. Meanwhile, INISg1 was only tested within limited experimental conditions, which cannot guarantee the robustness of INISg1 in various experimental conditions. Thus, this study aims to introduce variants of INISg1 and perform a thorough comparison of the original INISg1 and its variants. MethodsIn this study, based on the right Riemann sum (RR) and trapezoid rule (TR) of numerical integration, INISg1_RR and INISg1_TR are suggested. They are thoroughly compared with the original INISg1 using model-based simulations that offer us control of most of the experimental conditions, including integration time, β, and photon count rate. ResultsExcept for some extreme cases, INISg1 performed more robustly than INISg1_RR and INISg1_TR. However, in extreme conditions, variants of INISg1 performed better than INISg1. With the same condition, the signal processing speed of INISg1 was 1.63 and 1.98 times faster than INISg1_RR and INISg1_TR, respectively. ConclusionThis study shows that INISg1 is robust in most cases and the study can be a guide for researchers using INISg1 and its variants in different types of DCS applications.

Symplectic numerical integration for Hamiltonian stochastic differential equations with multiplicative Lévy noise in the sense of Marcus

In this paper, we propose a symplectic numerical integration method for a class of Hamiltonian stochastic differential equations with multiplicative Lévy noise in the sense of Marcus. We first construct a general symplectic Euler scheme for these equations, then we prove its convergence. In addition, we provide realizable numerical implementations for the proposed symplectic Euler scheme in detail. Some numerical experiments are conducted to demonstrate the effectiveness and superiority of the proposed method by the simulations of its orbits, Hamiltonian and convergence order over a long time interval. The results show the applicability of the methods considered.

Performance Analysis of Solar Tracking Systems by Five-Position Angles with a Single Axis and Dual Axis

This research presents an analysis of the five-position angle in both single-axis (one-axis tracking) and dual-axis (two-axis tracking) solar tracking systems. The study compares these tracking systems, where four solar panels move simultaneously, with a fixed solar panel system. The findings revealed that the five-position angle Sun-tracking technique resulted in lower energy consumption by the tracking mechanism than in the case of an all-time solar tracking system. The key component of the implemented system is a light-dependent resistor (LDR) sensor for controlling the motion of the motor for five positions on the vertical axis and horizontal axis, processed by a microcontroller to ensure the necessary solar tracking always moves in a perpendicular direction. According to the results, the voltage, current, and power increased with both one-axis and two-axis tracking compared to those of the fixed solar panel system under the same conditions. However, when evaluating the total energy with numerical integration methods, one-axis and two-axis provided 183.12 Wh and 199.79 Wh, respectively. Consequently, the energy production of the one-axis tracking system and the one-axis tracking system was found to be 16.71% and 24.97%, respectively, when compared to the fixed-axis system. Thus, the five-position angles of the sun-tracking technique resulted in lower energy consumption than is the case of an all-time solar tracking system.

Algorithm for automation of the method of testing statistical hypotheses according to the Pearson criterion by the means of MATLAB

The purpose of the work is to develop and improve the methods of computer modeling and digitalization of procedures for testing statistical hypotheses according to consent criteria for use in high-tech software and measuring complexes of automated control systems. In most cases that are important for practice, the analysis of the statistical properties of distributions requires multiple checks performed using numerical methods, which require the tools of powerful software and computing environments. Examples of such cases are the procedures for finding information subsets of features in regression and statistical analysis, as well as identifying parameter deviations in automated control systems. It is noted that the tasks of processing experimental data do not always fit into the framework of the theory of normal processes and have exclusively such hypotheses. To test such hypotheses, modern mathematical statistics has developed criteria based on specially developed probabilistic rules that distinguish between zero and alternative hypotheses. Moreover, if for the testable (zero) hypothesis the distribution of statistics is known exactly or approximately, then for the alternative it has a greater degree of uncertainty and it (the alternative) itself is not a complete negation of the null hypothesis. As the zero and alternative hypotheses move away from each other, the statistics of the consent criterion begins to take a larger absolute value than with the null hypothesis, which allows us to build a critical area on the basis of estimating the set of this statistics values. However, the use of analytical methods for statistical analysis to determine the discrepancy between empirical and hypothetical statistics in practice is associated with the use, as a rule, of complex computational calculation algorithms and the implementation of high requirements for the assessment accuracy, which they cannot provide. To solve this problem, it is proposed to use numerical methods of computer modeling and integration of probability density of distribution according to the Simpson algorithm using the tools of built-in functions of the Statistics Toolbox of the MATLAB environment. The proposed method for automation of statistical analysis of zero and alternative hypotheses is implemented on the example of applying the Pearson criterion to verify compliance with the normal distribution of two random data samples, one of which meets the normal distribution law. The obtained graphic and numerical results of computer modeling confirm the correspondence of one of the testable distributions to a given normal distribution according to the null hypothesis.

Quadratized Taylor series methods for ODE numerical integration

We focus on Taylor Series Methods (TSM) and Automatic Differentiation (AD) for the numerical solution of Ordinary Differential Equations (ODE) characterized by a vector field given by a finite composition of elementary and standard functions. We show that computational advantages are achieved if a kind of pre-processing said Exact Quadratization (EQ) is applied to the ODE before applying the TSM and the AD. In particular, when the ODE function is given by a formal polynomial (i.e. with real powers) of n variables and m monomials, the computational complexity required by our EQ based method for the calculation of the k-th order Taylor coefficient is O(k) whereas by using the existing AD methods it amounts to O(k2).

Some of the variables, some of the parameters, some of the times, with some physics known: Identification with partial information

Experimental data is often comprised of variables measured independently, at different sampling rates (non-uniform Δt between successive measurements); and at a specific time point only a subset of all variables may be sampled. Approaches to identifying dynamical systems from such data typically use interpolation, imputation or subsampling to reorganize or modify the training data prior to learning. Partial physical knowledge may also be available a priori (accurately or approximately), and data-driven techniques can complement this knowledge. Here we exploit neural network architectures based on numerical integration methods and a priori physical knowledge to identify the right-hand side of the underlying governing differential equations. Iterates of such neural-network models allow for learning from data sampled at arbitrary time points without data modification. Importantly, we integrate the network with available partial physical knowledge in “physics informed gray-boxes”; this enables learning unknown kinetic rates or microbial growth functions while simultaneously estimating experimental parameters.